How to account for this universal relationship?
For \(\theta _m=0\) , the whole RBC cell can be considered as an almost OB one, which is characterized by \(Ra\)  and the corresponding critical Rayleigh number \(Ra_{c,0}\)  for the onset of convection in this cell.
For \(\theta _m>0\) , only the lower layer can be treated as an OB cell.
The temperature at this layer's upper surface is \(T_m\)  and the corresponding Rayleigh number and critical Rayleigh number are \(Ra_\ell \)  and \(Ra_{c}\) , respectively.
From
2D OB DNS with \(Pr=10\)  [1]} we know that in the considered \(Ra\) -range, \(Nu\)  scales as \((Ra/Ra_{c,0})^\gamma \)  with \(\gamma \approx 0.27\) . Therefore
\(Nu(\theta _m)/Nu(0)=(Ra_\ell /Ra_c)^\gamma (Ra_{c,0}/Ra)^\gamma =(Ra_\ell /Ra)^\gamma (Ra_{c,0}/Ra_c)^\gamma \) .
Due to the reduced height and the temperature drop in the lower layer compared to the whole RBC cell (by almost \((1-\theta _m)\) , as a first approximation), the value of \(Ra_\ell /Ra\)  can be approximated as \((1-\theta _m)^4\) .
The critical Rayleigh numbers for the onset of convection can be approximated as
\(Ra_{c}\sim [1+(1-\theta _m)^2/\Gamma ^2]^2\)  and
\(Ra_{c,0}\sim [1+1/\Gamma ^2]^2\) 
(see more information in the Supplementary material), which altogether lead to
\(\hspace{-14.22636pt}\frac{Nu(\theta _m)}{Nu(0)}\approx \left[\frac{(1+\Gamma ^2)(1-\theta _m)^2}{\Gamma ^2+(1-\theta _m)^2}\right]^{2\gamma }=\left[\frac{5(1-\theta _m)^2}{4+(1-\theta _m)^2}\right]^{0.54}.\) 
